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submitted to ApJ

Large Magellanic Cloud Bump Cepheids: Probing the Stellar

Mass-Luminosity Relation

S. C. Keller

IGPP, L-413, LLNL, PO Box 505, Livermore, CA 94550

skeller@igpp.ucllnl.org

and

P. R. Wood

RSAA, Australian National University, Canberra A.C.T. 2600, Australia

wood@mso.anu.edu.au

ABSTRACT

We present the results of non-linear pulsation modelling of 20 bump Cepheids

in the LMC. By obtaining a optimal fit to the observed V, R MACHO lightcurves

we have placed tight constraints on stellar parameters of M , L, Teff and well as

quantities of distance modulus and reddening. We describe the mass-luminosity

relation for core-He burning for intermediate mass stars. The mass-luminosity

relation depends critically on the level of mixing within the stellar interior over the

course of the main-sequence lifetime. Our sample is significantly more luminous

than predicted by classical stellar evolutionary models that do not incorporate

extension to the convective core. Under the paradigm of convective core overshoot

our data implies Λc of 0.65±0.03l/Hp. We derive a LMC distance modulus of

18.55±0.02.

Subject headings: Cepheids:pulsation stellar:evolution

1. Introduction

Cepheids are classical distance indicators. Their tight conformity to a period-luminosity

relationship has made them the fundamental basis of the extra-galactic distance scale and

– 2 –

hence integral to observational cosmology. Ideally, we would like to have theoretical models

capable of accurately predicting the period-luminosity relation and its metallicity depen-

dence. The regularity of Cepheid pulsation provides a set of well defined observational

parameters with which to confront the predictions of theoretical models of stellar pulsations.

In this way, Cepheids provide us the ability to closely scrutinise the accuracy of input physics

within pulsation models.

Cepheid light curves display a variety of shapes and amplitudes that are period depen-

dent (the Hertzsprung (1926) progression). A feature of the lightcurves of some Cepheids is a

pronounced bump either preceding or following maximum. The bump arises from resonance

between the fundamental mode and second overtone. This resonance becomes particularly

prominent when the period ratio of these two modes (P0/P2=P02) is ∼ 2.

The bump enables us to break the degeneracy that exists between observable quanti-

ties; lightcurve shape, amplitude, period and the intrinsic properties; mass, luminosity and

temperature of the Cepheid. Such a technique was first proposed by Stobie (1969) and

was demonstrated by Wood, Arnold, & Sebo (1997)(hereafter Paper 1) in their non-linear

pulsation analysis of the LMC bump Cepheid HV 905.

Through the analysis of bump Cepheids we have a probe of the stellar mass-luminosity

(M-L) relation for core He-burning stars. The M-L relation depends critically on the size

of the central He core established (largely) during the course of the star’s main-sequence

lifetime. The size of the He core is in turn, determined by the extent of convection in the

vicinity of the convective core.

The treatment of convection remains the weakest point in our description of massive

stars. Ongoing debate focuses on the degree of extension to the convective core beyond that

predicted by standard, non-rotating stellar evolution models. Extension of the convective

core has traditionally been discussed in terms of convective core overshoot (CCO) in the

formalism of mixing-length theory. The CCO parameter, Λc, sets the height (as a fraction of

the pressure scale height) to which gas packets from the convective core rise into the formally

convectively stable region surrounding the core.

Mixing in the vicinity of the convective core produces a number of important evolu-

tionary changes that are expressed in a stellar population. It expands the amount of H

available to the core and hence extends the main-sequence lifetime. The star consequently

develops a more massive He core and the subsequent post-main-sequence evolution occurs

at a more rapid pace and at higher luminosities. That is, the M-L relation is significantly

more luminous than that of classical models.

Numerous studies have attempted to ascertain the efficiency of CCO from a theoret-

– 3 –

ical basis with results that range from negligible to substantial (see e.g. Bressan, Chiosi,

& Bertelli (1981)). An analytical approach appears limited given the complexity of the

phenomenon. Laboratory fluid dynamics shows that an understanding of convective mixing

requires a description of the turbulence field at all scales, a problem that will require detailed

hydrodynamical modelling.

Observations are required to ascertain the amount of CCO to apply in stellar evolu-

tionary models. Many studies have sought to do so through the study of young cluster

populations (most recently Barmina et al. (2002); Keller, Da Costa, & Bessell (2001)) and

from the broader field population (Beaulieu et al. 2001; Cordier et al. 2002) of the Magellanic

Clouds. Whilst large uncertainties exist in the derived values of Λc, the broad consensus of

these studies is the necessity of some level of CCO.

Another way of quantifying Λc is to use masses and luminosities of Cepheids. Dynamical

masses for Cepheids are presented by Evans et al. (1997, 1998) and Bohm-Vitense et al.

(1997a,b). Derived masses have considerable uncertainties but the combined sample (Evans

et al. 1998) indicates the necessity for some level of CCO.

This study aims to quantitatively establish the level of CCO by an examination of the

M-L relation of a sample of bump Cepheids from the LMC. A consistent result of pulsation

modelling is that the M-L relation for Cepheids is significantly brighter than predicted by

classical stellar evolution. The study of the bump Cepheid HV905 in Paper 1 found a

bump mass 29% lower than that required by evolutionary models without CCO. Recently,

Bono, Castellani, & Marconi (2002) applied non-linear modelling techniques that incorporate

turbulent convection to two LMC bump Cepheids. They found that an acceptable match

between model and observed lightcurves required a mass-luminosity relation in which stars

are ∼15% lower in mass than predicted by evolutionary models that neglect convective core

overshoot. Linear pulsation analyses (Sebo & Wood 1995; Kanbur & Simon 1994) similarly

require pulsation masses for Cepheids that are significantly lower than evolution masses.

2. Observations

Photometry for the LMC bump Cepheids considered here is taken from the MACHO

photometric database. Stars are only considered from the central bar region (the top 22

MACHO fields) in which standardised photometry exists. Magnitudes in the MACHO B and

R passbands have been converted to Kron-Cousins V and R using existing transofromations

described in Alcock et al. (1999). Photometric uncertainties are quoted as ±0.035 mag in

zero point and V −R colour. The observed Cepheids are listed in Table 1.

– 4 –

3. Model Details

Details of the non-linear pulsation code are given in Paper 1. The opacities have been

updated to OPAL 96 (Iglesias & Rogers 1996), supplemented at low temperatures by those

of Alexander & Ferguson (1994). Convective energy transport was included by means of

mixing-length theory with the assumption of a mixing length of 1.6 pressure scale heights.

A linear non-adiabatic code was used to derive the starting model for each simulation.

Models contained 460 mass points outside an inner radius of 0.3 R⊙. Transformation of L

and Teff into V and V −R of our observations was made through interpolation into a grid

of synthetically derived colours and bolometric corrections. The colors were computed for

the revised Kurucz (1993) fluxes used in Bessell, Castelli, & Plez (1998) and described in

more detail in Castelli (1999). Magnitudes were computed through energy integration using

the passbands of Bessell (1990). We computed non-linear models at a fixed composition of

Y=0.27 and Z=0.008 found for young objects in the LMC (Russell & Bessell 1989).

In contrast to Bono, Castellani, & Marconi (2002), our method uses only stellar pul-

sation and stellar atmosphere theory, we do not make recourse to existing mass-luminosity

(M-L) relations. Once abundance is assumed three parameters, M , L and Teff remain to

characterise the the pulsating envelope of each Cepheid. Thus three conditions are required

to determine these quantities.

The first condition is that the fundamental pulsation period of the starting linear models

must satisfy the observed period of the Cepheid. The other two conditions come from

fitting non-linear model lightcurves to the observations. The two parameters we chose as

independent variables for the lightcurve fit were Teff and P02, the ratio of the fundamental to

second overtone periods. The amplitude of pulsation is dependent on the star’s temperature

relative to the blue edge of the IS. Hence the amplitude of pulsation is a strong constraint

on Teff . The phase of the bump is dependent on P02 and furthermore, is independent of the

pulsation amplitude (Simon & Lee 1981).

To commence the modelling process values of Teff and P02 were specified and parameters

L and M were iterated until the required linear period and P02 were produced. Once model

parameters were determined, the static model was perturbed with the eigenfunction of the

linear adiabatic fundamental mode. The perturbed model was run until the kinetic energy

of the pulsation reached a limit cycle.

Our models incorporate convective energy transfer through the mixing length approxi-

mation. This is known to be only a partial description of the internal physics in a Cepheid

atmosphere. In particular, at cooler temperatures as the convective regions become larger

and the dynamical timescale becomes a significant fraction of the pulsation period our models

– 5 –

are expected to become increasingly divergent. This is a well known shortcoming of models

that implement the mixing-length approximation. Whilst our models can match the blue

edge of the instability strip (IS) they can not reproduce its red edge. In the vicinity of the

red edge the amplitude of pulsation is too high, a feature that can not be circumvented by

modification of artificial viscosity parameters. To produce a physical red edge an additional

form of energy dissipation is required. Convective processes are the likely cause of this.

Yecko, Kollath, & Buchler (1998) shows that models with a parametrised formulation of

turbulent convective mixing are able to match the fundamental and first overtone instability

strips through a fine tuning of parameters.

Yecko et al. consider the case of a 5M⊙ star modeled both with mixing-length approxi-

mation and with turbulent convection. Consideration of the model growth rates (their figure

11) shows insignificant differences over the bluest 1/4 of the IS, becoming increasingly diver-

gent thereafter. In order to avoid as much as possible the short comings of the mixing-length

approach we have sought bump Cepheids close to the blue edge of the IS (see Fig. 1).

4. Results

In figures 2 & 3 we show the effects of variation of the two parameters Teff and P02.

As we change P02 we modify the phase at which the bump is located. Similarly, as Teff is

varied the amplitude is changed. The best fit to the observed light curve is shown in the

central panel. Here for the purpose of illustration we show five models widely separated

in parameter space. In the determination of the best model, however, we use a iterative

chi-squared minimisation technique.

The offset of the model MV and observed V light curves gives the apparent distance

modulus. Having obtained a model that matches the V light curve of each bump Cepheid, the

observed V -R colour curve was shifted onto the model intrinsic V -R colour curve. The shift

required to do so is the colour excess, EV −R, which can be converted to visual absorption,

AV , using a standard reddening curve (AV =1.78E(V − R) was used). The true distance

modulus can then be obtained from the apparent distance modulus. The best fit parameters

for the Cepheids in our sample are given in Table 2.

Limits of internal accuracy in the determination of the best model solution arises from

cycle-to-cycle variations in the model output. By monitoring on the order of 30 cycles we can

quantify the uncertainty introduced by these model variations. The errors in the determined

parameters are dominated by the uncertainty in P02. This becomes problematic towards

lower masses as the bump amplitude diminishes. Systematic uncertainties dominate the total

– 6 –

uncertainty however. The quoted photometric uncertainties for the MACHO photometry

account for ±0.1M⊙ in mass and ±0.002 dex in luminosity. Error bars shown in figure 4 are

the quadrature sum of systematic and internal uncertainties.

As stated above, a value of metallicity is adopted in our analysis. It is important to

remark on the effects that varying this metallicity has on our models. An exploration of this

has been presented by us in Paper 1. In this work, models were additionally constructed for

solar and SMC metallicities. It was found that the models cannot be used to distinguish

the abundance of the Cepheid with any meaningful uncertainty. If the abundance of our

sample is assumed to lie in the range Z=0.006-0.01 and Y=0.25-0.29 the resulting error in

the derived parameters from the uncertainty due to abundance is of similar magnitude as

that reported from P02 and Teff .

A less constrainable systematic effect may arise from our treatment of convection. We

note the work of Buchler et al. (1996) and Feuchtinger, Buchler & Kollath (2000) who find

that the effects of convection are important even in the vicinity of the blue edge. Whilst a

large degree of freedom is available in the internal parameters of turbulent convection models

the magnitude of this possible systematic effect remains uncertain.

4.1. Reddenings and Distance Modulus

Another independent prediction of the bump Cepheid models is the reddening to each

object. In figure 5 we present the histogram of determined reddenings. Our derived mean

reddening is 0.08.

A number of studies of line-of-sight redddening to LMC stars have been presented in

the literature. Bessell (1991) showed that the historically low values for reddening to the

LMC (< E(B−V )>∼0.03) were too low and used the Johnson & Morgan reddening-free Q

index and spectroscopic temperatures to show that the <E(B−V )>∼0.12. Harris, Zaritsky,

& Thompson (1997) use Q to form a histogram of E(B−V ) from a 2.8◦2 region of the LMC.

They find a mean reddening of 0.20 with a non-gaussian tail extending to higher E(B − V ).

Larsen, Clausen, & Storm (2000) report <E(B−V )>= 0.085. Zaritsky (1999) revisits the

data of Harris et al. and reports a strong dependence of <E(B−V )> on the spectral type of

objects used to define it. Namely, low extinctions result from red clump stars, high extinction

from OB stars. Zaritsky proposes that is the result of a larger scale height for older stars,

placing them statistically in lower reddening regions than the OB stars that reside in the

dusty disk. With this background we find that our reddenings are applicable for objects

within the LMC.

– 7 –

The mean LMC distance modulus shown in Figure 5 is 18.55±0.02. This in good

agreement with recent analysis (18.57±0.14 from the compilation of Gibson 2000).

The consistency of our derived reddening and distance modulus with previous studies

provides a validation to the input of our model. It means that the results we present here

will be consistent with results derived from linear pulsation models. Linear pulsation models

that utilize the observed colours, reddenings, apparent luminosity and distance modulus by

Sebo & Wood (1995) do indeed show pulsation masses for LMC Cepheids that are similarly

lower than evolution masses. This gives us confidence in the consistency of the linear and

non-linear pulsation theory as well as the bolometric corrections and colour transformations.

4.2. The Mass-Luminosity Relation

By modelling the bump Cepheid sample we have independently determined both L and

M for each object. We use these values to construct the mass-luminosity relation for core-He

burning stars in the LMC. Figure 4 shows the mass-luminosity relationship described by our

sample. Overlaid are the M-L relations (due to Fagotto et al. (1994) and Bressan (2001))

for three levels of convective core overshoot efficiency with Z=0.008 and Y=0.25.

The data are significantly more luminous than the predictions of standard “mild” con-

vective core overshoot (i.e. Λc = 0.5). Figure 4 recommends a degree of convective core

overshoot of Λc = 0.65 ± 0.03l/Hp. Put another way, our results favour a mass 19.5±1.0%

lower than classical evolutionary models.

The problem of reconciling mass determinations from the various techniques available

has been a problem that has plagued the field. The many phases of debate, and their

convergence, have been extensively discussed in the literature (see e.g. Cox (1980)). The

longest standing of these, the bump and beat Cepheid mass discrepancy, has to a large part

been resolved by Moskalik, Buchler, & Marom (1992) through the introduction of OPAL

opacities. The discrepancy between pulsation and evolutionary mass has not been removed

by improvements in input physics.

One possibility that has been suggested by Bono, Castellani, & Marconi (2002) is that

mass loss is responsible for the reduction in mass. As opposed to the evolutionary models used

in the study of Bono et al. that neglect mass loss, the models shown in figure 4 incorporate the

mass loss prescription of de Jager et al. (1988). This is largely responsible for the curvature

of the M-L relation towards higher masses. Furthermore, mass loss during the Cepheid phase

does not appear to be enhanced from that expected from the de Jager description (Deasy

1988). We conclude that mass loss alone can not explain the Cepheid mass discrepancy.

– 8 –

One Cepheid (MACHO 2.4661.3597: HV 905) has been the target of previous analysis

in Paper 1. However, the current work uses updated OPAL opacities and improved low tem-

perature opacities from Alexander & Ferguson (1994), and replaces bolometric corrections

of Kurucz (1993) with those of Castelli (1999). The results from this previous work were

M=5.15±0.35M⊙ and log(L/L⊙)=3.69±0.01 (uncertainties determined post-fact from the

details presented in Paper 1). Here we find M=5.62±0.22M⊙ and log(L/L⊙)=3.701±0.004.

In their description of their non-linear pulsation models that incorporate turbulent con-

vection, Bono, Marconi, & Stellingwerf (1999) compare their model output for HV 905 with

that of Paper 1. They use the mass, luminosity, Teff and abundance in Paper 1 and re-

trieve a limit cycle lightcurve that is very similar in shape to that of Paper 1. In addition,

the P02 differs by only 1% and the luminosity amplitude differs by 0.06-0.08 mag from that

presented in Paper 1. This gives us confidence that our study is unaffected by our treatment

of convection under the mixing length approximation.

Future refinement of the technique described here is possible through the comparison of

observed radial velocity variations of bump Cepheids to those predicted by our model. This

has the benefit of limiting systematic errors. At present, the accuracy of our technique is

limited by systematic photometric uncertainties (in particular transformations of MACHO

bandpasses and zeropoints). The radial velocity curve is a more robust output of the non-

linear pulsation model. Radial velocities are more closely related to the dynamical processes

within the stellar atmosphere than that of photometry which is more affected by temperature

changes than dynamical effects. Work is currently underway by us to use radial velocity

curves for bump Cepheids as a stringent test on our pulsation models. Models should be

able to accurately reproduce all observational constraints (period, shapes of light and velocity

curves). Such a critical examination of the dynamics of Cepheid pulsation offers the potential

of further model refinement.

We note that demonstrating the convective core is of greater extent than the core defined

by the Schwarzschild criterion does not determine its causation. Recent studies have shown

that rotation can provide a natural way to bring about increased internal mixing and hence

a larger core (Heger, Langer, & Woosley 2000; Meynet & Maeder 2000).

Attention to rotational mixing has been promoted by the findings of chemical abundance

studies of A supergiants (Venn 1995), B supergiants (Dufton et al. 2000), and main-sequence

B stars (Daflon et al. 2001; Gies & Lambert 1992) which imply a broad range of enhance-

ments. This behavior is interpreted as arising from self-contamination with CN-cycled ma-

terial prior to the first dredge-up. Such surface modifications can not be achieved through

CCO.

– 9 –

Furthermore, the CNO abundances for a set of 10 SMC A supergiants have been studied

by Venn (1999). These stars show a large range in N abundance enhancements ranging from

negligible to levels greater than first dredge-up. The magnitude of enhancement is much

greater than that found for a similar sample of Galactic A supergiants (Venn 1995) which

suggests a possible metallicity effect in the underlying physics of internal mixing.

Bump Cepheids of the SMC offer the opportunity to investigate the metallicity depen-

dence of internal mixing. A greater level of internal mixing such as would be produced by

generally higher stellar rotation velocities in the SMC (Venn 1999) should result in a clear

modification of the M-L relation. Work is underway to look for such effects.

5. Conclusions

In this study we have applied non-linear pulsation models to match the observed light

and colour curves of 20 LMC bump Cepheids. We have been able to place tight constraints

on the stellar parameters mass, luminosity and effective temperature as well as individual

reddenings. We have described the mass-luminosity relation over the mass range of bump

Cepheids. We find our sample is ∼20% more luminous for their mass predicted by stel-

lar evolution models that do not incorporate extension to the convective core. Under the

paradigm of convective core overshoot this amounts to Λc of 0.65±0.03l/Hp. The models

yield an LMC distance modulus of 18.55±0.02.

We thank A. Bressan et al. for providing us with unpublished evolutionary models for

Λc=1.0. We thank our referee Dr. Z. Kollath for his comments and personal calculations

regarding systematic effects due to turbulent convection. Work performed by SCK was

performed under the auspices of the U.S. Department of Energy, National Nuclear Security

Administration by the University of California, Lawrence Livermore National Laboratory

under contract No. W-7405-Eng-47.

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This preprint was prepared with the AAS LATEX macros v5.0.

– 13 –

Fig. 1.— The V , V −R colour-magnitude diagram for fundamental Cepheids in the MACHO

catalogue. The bump Cepheids of the present study are shown in bold. Overlaid are the blue

and red edge of the instability strip described by Chiosi, Wood & Capitano (1993) assuming

E(B−V )=0.12

– 14 –

Fig. 2.— Model fits for MACHO 2.4661.3597. The numbers in each panel are: the period

ratio P02 and log Teff in the upper part, and M/M⊙, log L/L⊙, and the true distance modulus

in the lower part. Each panel contains light and color curves (dereddened). Observations

are shown as dots and models as lines. The panels in the vertical section show the effect of

changing Teff , this affects the amplitude (in the upper panel Teff is too large, the resulting

amplitude is too low). The horizontal section shows the effect of a changing P02 this affects

the phase of the bump (in the left panel P02 is too small, the phase of the bump is too

“late”). The central panel is the best model.

– 15 –

Fig. 3.— As in Fig. 2, model fits for MACHO 79.5139.13.

– 16 –

Fig. 4.— The mass-luminosity relation for the present sample of 20 bump Cepheids. Error

bars are as discussed in the text. Overlaid are the M-L relations for core-He burning stars

from Fagotto et al. (1994) and Bressan (2001) for three assumptions of the efficiency of

convective core overshoot and Z=0.008 and Y=0.25. The solid line represent the best match

to the data (Λc=0.65 l/Hp).

– 17 –

0 0.05 0.1 0.15 0.20

2

4

6

8

E(B-V)

18.3 18.4 18.5 18.6 18.70

2

4

6

8

Distance modulus

Fig. 5.— Histograms of the derived reddenings (top) and distance moduli (bottom) for the

20 Cepheids of our sample.

– 18 –

Table 1: The selected MACHO bump Cepheid sample

MACHO star id RA (J2000) Dec (J2000) < V > < V −R > P [d]

1.3441.15 05 01 52.0 -69 23 22 14.45 0.37 10.4136

1.3692.17 05 02 51.4 -68 47 06 14.53 0.39 10.8552

1.3812.15 05 03 57.3 -68 50 25 14.61 0.39 9.7118

1.4048.6 05 05 08.8 -69 15 12 14.77 0.36 7.7070

1.4054.15 05 05 42.1 -68 51 06 14.93 0.37 7.3953

2.4661.3597 05 09 16.0 -68 44 30 14.32 0.39 11.85911

6.6456.4346 05 20 23.1 -70 02 33 15.16 0.36 6.4816

9.4636.3 05 09 04.5 -70 21 55 14.16 0.41 13.6315

9.5240.10 05 13 10.1 -70 26 47 15.11 0.38 7.3695

9.5608.11 05 15 04.7 -70 07 11 14.81 0.35 7.0693

18.2842.11 04 57 50.2 -68 59 23 14.83 0.38 8.8311

19.4303.317 05 06 39.8 -68 25 13 14.65 0.37 8.7133

19.4792.10 05 09 36.9 -68 02 44 14.96 0.36 6.8628

77.7670.919 05 27 55.1 -69 48 05 14.85 0.34 7.4423

77.7189.11 05 24 33.3 -69 36 41 14.73 0.39 7.7712

78.6581.13 05 20 56.0 -69 48 19 14.97 0.36 6.9302

79.4657.3939 05 08 49.4 -68 59 59 14.23 0.43 13.8793

79.4778.9 05 09 56.3 -68 59 41 14.56 0.33 8.1868

79.5139.13 05 11 53.1 -69 06 49 14.59 0.38 8.7716

79.5143.16 05 12 18.8 -68 52 46 14.61 0.34 8.2105

– 19 –

Table 2: The details of our sample

MACHO star id P [d] P02 log(Teff) E(B − V ) µobj log(L/L⊙) M/M⊙

1.3441.15 10.4136 2.005 3.757 0.078 18.554 3.649 5.698

1.3692.17 10.8552 2.012 3.756 0.072 18.622 3.653 5.793

1.3812.15 9.7118 1.980 3.762 0.076 18.569 3.639 5.778

1.4048.6 7.7070 1.938 3.765 0.060 18.626 3.455 5.093

1.4054.15 7.3953 1.940 3.767 0.065 18.563 3.472 4.988

2.4661.3597 11.8591 2.030 3.755 0.088 18.548 3.701 5.623

6.6456.4346 6.4816 1.940 3.770 0.093 18.665 3.267 4.368

9.4636.3 13.6315 2.035 3.756 0.078 18.478 3.851 6.501

9.5240.10 7.3695 1.952 3.767 0.082 18.477 3.439 4.606

9.5608.11 7.0693 1.92 3.765 0.155 18.735 3.460 5.249

18.2842.11 8.8311 1.973 3.762 0.049 18.431 3.563 5.317

19.4792.10 6.8628 1.924 3.766 0.051 18.548 3.424 4.886

19.4303.317 8.7133 1.958 3.763 0.068 18.564 3.597 5.663

77.7670.919 7.4423 1.936 3.765 0.078 18.543 3.491 5.244

77.7189.11 7.7712 1.940 3.764 0.092 18.501 3.509 5.355

78.6581.13 6.9302 1.930 3.764 0.084 18.540 3.385 4.920

79.4657.3939 13.8793 2.032 3.758 0.112 18.528 3.815 6.742

79.4778.9 8.1868 1.953 3.762 0.071 18.643 3.527 5.364

79.5139.13 8.7716 1.968 3.763 0.068 18.509 3.570 5.417

79.5143.16 8.2105 1.95 3.763 0.114 18.571 3.540 5.465